Problem 103 - Chemistry The Central Science In Si Units

Question

Both dimethylhydrazine, \(\left(\mathrm{CH}_{3}\right)_{2} \mathrm{NNH}_{2},\) and methylhydrazine, \(\mathrm{CH}_{3} \mathrm{NHNH}_{2},\) have been used as rocket fuels. When dinitrogen tetroxide \(\left(\mathrm{N}_{2} \mathrm{O}_{4}\right)\) is used as the oxidizer, the products are \(\mathrm{H}_{2} \mathrm{O}, \mathrm{CO}_{2},\) and \(\mathrm{N}_{2} .\) If the thrust of the rocket depends on the volume of the products produced, which of the substituted hydrazines produces a greater thrust per gram total mass of oxidizer plus fuel? (Assume that both fuels generate the same temperature and that \(\mathrm{H}_{2} \mathrm{O}(g)\) is formed.)

Step-by-Step Solution

Answer

Methylhydrazine produces greater thrust per gram than dimethylhydrazine.

Step 1: Write Balanced Chemical Equations

Firstly, we need to balance the chemical equations for each fuel reacting with dinitrogen tetroxide. \(\text{For } (\mathrm{CH}_{3})_{2} \mathrm{NNH}_{2}:\)\[ \mathrm{2(CH_3)_2NNH_2 + 3N_2O_4 \rightarrow 4NO_2 + 2N_2 + 4CO_2 + 6H_2O} \]\(\text{For } \mathrm{CH_3NHNH_2}:\)\[ \mathrm{2CH_3NHNH_2 + 2N_2O_4 \rightarrow 4NO_2 + 3N_2 + 2CO_2 + 6H_2O} \] Ensure both equations are balanced correctly with respect to each of the reactant and product atoms.

Step 2: Determine the Volume of Gaseous Products

Calculate the total moles of gaseous products formed from each reaction, considering that \(\mathrm{H_2O}\) is in gaseous form. Using the balanced equations:- For \( (\mathrm{CH}_3)_2\mathrm{NNH}_2 \), the total moles of gases: \( 4\mathrm{NO_2} + 2\mathrm{N_2} + 4\mathrm{CO_2} + 6\mathrm{H_2O} = 16 \) moles.- For \( \mathrm{CH_3NHNH_2} \), the total moles of gases: \( 4\mathrm{NO_2} + 3\mathrm{N_2} + 2\mathrm{CO_2} + 6\mathrm{H_2O} = 15 \) moles.

Step 3: Calculate the Mass of Reactants

Calculate the total molar mass of the reactants for each reaction.- For \( (\mathrm{CH}_3)_2\mathrm{NNH}_2 \):\(\mathrm{C_2H_8N_2 = 60.1 \ g/mol}\), \( \mathrm{N_2O_4 = 92.0 \ g/mol} \). Total mass: \((2 \times 60.1) + (3 \times 92.0) = 396.2 \ g \).- For \( \mathrm{CH_3NHNH_2} \):\(\mathrm{CH_6N_2 = 46.1 \ g/mol}\), \(\mathrm{N_2O_4 = 92.0 \ g/mol} \). Total mass: \((2 \times 46.1) + (2 \times 92.0) = 276.2 \ g \).

Step 4: Calculate the Thrust per Gram

Determine which reaction produces more thrust by calculating the ratio of total moles of gaseous products to total mass of reactants for each fuel.- For \( (\mathrm{CH}_3)_2\mathrm{NNH}_2 \): \( \frac{16}{396.2} = 0.0404 \) moles/g.- For \( \mathrm{CH_3NHNH_2} \): \( \frac{15}{276.2} = 0.0543 \) moles/g. Compare the ratios to determine which fuel produces greater moles of gases per gram of reactant.

Key Concepts

Chemical Reaction Balancing

In chemistry, balancing chemical equations is essential because it ensures that the Law of Conservation of Mass is observed. This law states that matter cannot be created or destroyed in a chemical reaction.
Here's how it works: When both dimethylhydrazine and methylhydrazine are used as rocket fuels with dinitrogen tetroxide as an oxidizer, they react to form products such as \( ext{NO}_2 \), \( ext{N}_2 \), \( ext{CO}_2 \), and \( ext{H}_2 ext{O} \). To illustrate this, one needs to make sure the same number of each type of atom appears on both the reactant and product sides of the equation.

• For dimethylhydrazine: \( ext{2(CH}_3)_2 ext{NNH}_2 + 3 ext{N}_2 ext{O}_4 ightarrow 4 ext{NO}_2 + 2 ext{N}_2 + 4 ext{CO}_2 + 6 ext{H}_2 ext{O} \).
• For methylhydrazine: \(2 ext{CH}_3 ext{NHNH}_2 + 2 ext{N}_2 ext{O}_4 ightarrow 4 ext{NO}_2 + 3 ext{N}_2 + 2 ext{CO}_2 + 6 ext{H}_2 ext{O} \).

Balancing involves adjusting coefficients next to chemical formulas. It's like solving a puzzle where you have to figure out the right pieces for each side to be equal. After achieving this balance, the reaction is set to proceed under consistent and predictable conditions.

Oxidizer

An oxidizer is a critical component in rocket propellants, serving as a chemical agent that supplies the oxygen needed for fuel combustion. Without it, the rocket won't ignite and produce thrust.
In our exercise, dinitrogen tetroxide (\( ext{N}_2 ext{O}_4 \)) serves as the oxidizer. It reacts with the rocket fuels in question to generate sufficient energy and combustion needed for propulsion.

• This compound has the potential advantage of being stable at room temperature, making it more convenient for storage and use in rockets.
• During the combustion process, the oxidizer dissolves, releasing oxygen which then reacts with the hydrazine derivative fuels. This reaction produces gaseous substances essential for creating thrust.

The proper choice and balance of the oxidizer with the corresponding fuel impacts the efficiency and power of the propulsion system. Understanding how oxidizers function is essential in aerospace and chemical engineering, highlighting their central role in fuel systems.

Gaseous Products

Gaseous products in a chemical reaction within rocket propulsion systems play a prominent role. They directly influence the thrust generated by the rocket.
When dinitrogen tetroxide (\( ext{N}_2 ext{O}_4 \)) is used with dimethylhydrazine or methylhydrazine, a variety of gases like nitrogen dioxide (\( ext{NO}_2 \)), nitrogen (\( ext{N}_2 \)), carbon dioxide (\( ext{CO}_2 \)), and water vapor (\( ext{H}_2 ext{O} \)) are formed.

• For the dimethylhydrazine setup, 16 moles of gaseous products are generated, consisting of multiple gas compounds.
• On the other hand, the methylhydrazine reaction yields 15 moles of gases.

These gaseous products expand rapidly when heated, which is why they are crucial in the propulsion process. Their formation shows how the chemistry involved in fuel-oxidizer reactions is designed not only to sustain combustion but also to produce substances that expand and generate motion.

Thrust Calculation

Thrust is a measure of how effectively a rocket can propel itself. It's determined by examining the gases produced by the chemical reaction and how their mass compares to the mass of the fuel and oxidizer.
To determine which fuel is more efficient in terms of thrust per gram, consider the ratio of moles of gaseous products to total mass of reactants.

• Dimethylhydrazine produces a ratio of 0.0404 moles/g, calculated from 16 moles of gases over 396.2 grams of reactants.
• Methylhydrazine exceeds this with 0.0543 moles/g, derived from 15 moles of gases over 276.2 grams of reactants.

This tells us that methylhydrazine is more efficient in terms of creating thrust per gram of material. The method leverages both chemistry and mathematics to assess the effectiveness of fuels for aerospace applications, demonstrating how precise calculations can achieve superior rocket performance.